DISCIPLINE-ERR-IAN: Mary Crangle, an arbitrator charged with disciplining teachers, has let many off with fines or other wrist slaps.

Mary Crangle let teachers wangle.

The arbitrator refused to fire a Brooklyn teacher whose second-grader went on a bathroom break and wound up wandering Times Square for hours. She slapped the educator with a $1,000 fine.

The 7-year-old boy left Faye Dolloway’s class at PS 174 in Brownsville at 10:45 a.m. and got on a train. She didn’t realize he was missing until 1:20 p.m. Cops found him at 5:30 roaming the Toys “R” Us.

“I conclude there is no just cause for termination,” Crangle said. “[Dolloway] clearly understands her obligations as a teacher and I find her to be genuinely remorseful.”

The case is one of many in which Crangle and other high-paid hearing officers gave out light punishments to educators the city Department of Education wanted canned.

Of 76 educators who went before arbitrators this year, only 29 were fired — leaving 61 percent cleared to return to the classroom.

The hearing officers, jointly picked by the DOE and the United Federation of Teachers, preside over costly trials that can drag on for months. All of the current arbitrators are lawyers and get $1,400 a day plus travel expenses, paid by the state. The DOE pays its lawyers and staff to conduct the cases.

“It’s an overly bureaucratic, expensive and difficult system,” said Marcus Winters of the Manhattan Institute. “It’s really disheartening how difficult it is to do something tangible with teachers who don’t belong in the classroom.”

The DOE’s record on sexual misconduct cases is particularly troubling.

Since July 2008, the department has brought 100 charges for sexual misconduct and sought those teachers’ termination, a department official said. Of those, arbitrators allowed 74 to remain in schools and fired only 26.

In the 76 rulings issued for teachers this year, eight were exonerated. Arbitrators found teachers guilty of some misconduct or incompetence in 68 rulings but let 38 off with a fine, suspension or reprimand.

Critics blast the DOE’s discipline unit for failing to prove cases.

“They don’t know what they’re doing and don’t know how to do it,” a former insider charged.

The DOE blamed politics.

“Arbitrators frequently order more lenient punishment because they try to keep both the UFT and DOE happy,” said spokeswoman Connie Pankratz. “Reforming this system by allowing school districts more say in whether a teacher should be fired is an important part of being able to provide every student an excellent teacher.”

Among the recent rulings:

* Altagracia Liciaga, former principal at the Multicultural HS in Brooklyn, made five teens miss an entire school day to fetch file cabinets in Manhattan. The students rode in a U-Haul’s cargo hold. But hearing officer Martin Scheinman found Liciaga “genuinely upset and contrite.” He ordered a five-month suspension and a course on interpersonal skills. Liciaga’s salary is $149,213, but she’s not assigned to a school.

* Patricia Dawson, a teacher at the HS of Economics and Finance in Manhattan, berated students on Facebook as “suicide-inducing” and wrote, “I need a gun.” Hearing officer Joshua Javits called the comments “outrageous” but found her “not irredeemable.” He ordered a $15,000 fine and a course on “appropriate boundaries.”

* Deborah Rykman, a teacher at Thomas Giordano MS in The Bronx, yelled at her principal, was excessively late and absent and let kids sleep, play video games and throw snowballs in class. Hearing officer Stephen Bluth ordered a $1,500 fine and a workshop.

* Felicia Alterescu, a teacher at PS 79 in Queens, called in sick to attend a family reunion and failed to report two arrests. Crangle imposed a $7,000 fine. “I find her length of service to be a mitigating factor,” Crangle said of Alterescu’s 20 years in city schools. “I have decided to afford [her] the benefit of another opportunity.”

Wrist-slap happy

CASE 1

* Teacher: Pamela Garrison, Aspire Preparatory MS, Bronx

* Case: Allegedly gave calculators to 7th-grade students for a state math exam, even after a student told her they weren’t allowed.

* Case: Allegedly told a female colleague, “I have a picture of my brother’s penis.” Suggestively put his pinky in his mouth but argued he was just putting dentures in.

* Outcome: 6-month suspension, anger management

CASE 3

* Teacher: Lonwen Martins, PS 41, Brooklyn

* Case: Allegedly failed to supervise 6-year-olds at Adventurer’s amusement park in Bath Beach and left the park without a girl. Didn’t immediately notify the principal or turn around when she discovered the girl missing.